{
  "subject": "Graffiti girth>=5 campaign: 284 refutation verification + 290 candidate proof",
  "date": "2026-07-23",
  "statement_sources": {
    "wow_scan": "graffiti-292-artifacts/wow-july2004.pdf, PDF pages 79-80 (printed 79-80)",
    "transcription_method": "vision-model verbatim transcription of 5x rendered crops (wow-p79-bot.png, wow-p80-top.png)",
    "statements": {
      "284": "If girth is >= 5 then the minimum dual degree <= - the smallest eigenvalue of distance matrix.",
      "290": "If girth is >= 5 then the - 2nd smallest eigenvalue <= size/meangravity.",
      "292": "If girth is >= 5 then the minimum positive eigenvalue <= n/meangravity. [re-confirmed]",
      "295": "If girth is >= 5 then the number of positive eigenvalues of the distance matrix <= n/meangravity. [re-confirmed]"
    },
    "definition_risks": [
      "290 'eigenvalue' read as adjacency (Wall says 'of Laplacian' explicitly when Laplacian is meant: items 286/287/297); proof covers max|lambda| so any adjacency reading is subsumed",
      "gravity = Brewster/WotW page-52 definition, mean over all n^2 entries; R-C erratum documents that A-H's variant definition trivializes 290"
    ]
  },
  "claims": [
    {
      "id": "K1",
      "claim": "Graffiti 284 is FALSE; Hoffman-Singleton violates it with exact margin 3",
      "type": "exact computation (integer-only chain)",
      "evidence": "verify_284_hoffman_singleton_exact.py (in the Graffiti 284 archive) + independent_check_284_290.py",
      "priority": "counterexample first found by Capy Build run 2026-07-22 (tokiwa.space CE10; X posts 2026-07-23); our contribution is the exact verification"
    },
    {
      "id": "K2",
      "claim": "A^2+A-6I=J for HS; D=2(J-I)-A; spectrum of D = {91^1, 1^21, (-4)^28}",
      "type": "proved + integer-verified",
      "evidence": "verify_284_hoffman_singleton_exact.py (in the Graffiti 284 archive)"
    },
    {
      "id": "K3",
      "claim": "Graffiti 290 is TRUE: -lambda_{n-1}(A) <= m/mean_Gr for all connected girth>=5 graphs",
      "type": "proved (candidate first proof)",
      "evidence": "manuscript-graffiti-290.md; chain = 292 Lemmas 1-3 + terminal quartic"
    },
    {
      "id": "K4",
      "claim": "32*(2n(n-1)^2-(1+t)^3) = (t+1)^2 (t^4-2t^3+4t^2-38t-29); shift t=u+5 gives all-positive coefficients [1,18,124,352,256]",
      "type": "symbolically verified",
      "evidence": "verify_290_symbolic.py output"
    },
    {
      "id": "K5",
      "claim": "All 34 connected girth>=5 graphs with 2<=n<=7 satisfy 290 (exact rational gravity + Sturm root counts)",
      "type": "exact exhaustive computation",
      "evidence": "verify_290_small_cases.py output"
    },
    {
      "id": "K6",
      "claim": "Independent reimplementation (different constructions, cages via LCF, random trees, pruned random graphs) finds no violation of 290 and reproduces the 284 refutation",
      "type": "independent reimplementation",
      "evidence": "independent_check_284_290.py output"
    },
    {
      "id": "K7",
      "claim": "290 proof unrecorded in indexed literature (R-C 2024 open; A-H survey adjacency-side absent; BDF 1995 predates and R-C still list open)",
      "type": "bounded priority search",
      "evidence": "web searches 2026-07-23, self-echo (tokiwa/agnt.gg) excluded"
    },
    {
      "id": "K8",
      "claim": "Girth >= 5 and n >= 2: sum of gravity entries <= (2m)^2/n, so both readings of the mean gravity are <= 1; every eigenvalue has |lambda| <= Delta <= m <= m/mean",
      "type": "proved; exact checks on 345 graphs",
      "evidence": "paper section 2.5; battery_290.py and battery_290_output.txt"
    },
    {
      "id": "K9",
      "claim": "Graffiti 290 holds for both readings of the mean",
      "type": "formally verified (Lean 4 kernel)",
      "evidence": "lean/Graffiti/Graffiti290.lean (agnt-gg/graffiti-lean commit cceea87d0c42d56701ec3dea4f647b8a4bcf0309): graffiti_290, graffiti_290.variants.off_diagonal_mean, graffiti_290.variants.abs_le; Lean v4.33.1, Mathlib v4.33.1; #print axioms: propext, Classical.choice, Quot.sound"
    }
  ],
  "open_in_cluster": {
    "295": "OPEN. Statement re-confirmed. Analysis: RHS = n/mean_Gr >= 4n(n-1)/(1+sqrt(4n-3))^2 ~ n - O(sqrt(n)); LHS = n_+(D) <= n - n_-(D) <= n - diam(G) (geodesic-path principal submatrix + Graham-Pollack + Cauchy interlacing). Gap: need n_+(D) <= n - ~sqrt(n) or a sharper gravity bound using the C4-free codegree structure (only 2m ordered pairs at distance 1; W3 = 2*sum_E d(u)d(v) bound needed). Moore-graph checks: HS n_+(D)=22 vs RHS 89.3; hypothetical 57-Moore n_+(D)=1730 vs RHS ~6390. Named worst margin: Heawood 7 vs 38.6."
  }
}